Chapter 12

Toss a fair coin 150 times. Use the central limit theorem and the histogram correction to find an approximation for the probability that the number of heads is at least \(70 .\)

The following table contains the scores of 25 students on a certain exam: $$\begin{array}{rrrrr}7 & 8 & 8 & 3 & 2 \\ 5 & 6 & 9 & 10 & 6 \\ 8 & 8 & 7 & 6 & 9 \\ 10 & 4 & 4 & 8 & 6 \\ 9 & 10 & 5 & 5 & 8\end{array}$$ (a) Find the relative frequency distribution. (b) Compute the average value by (i) averaging the values in the table directly and (ii) using the relative frequency distribution obtained in (a).

Assume that a quantitative character is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Determine what fraction of the population falls into the given interval. \((-\infty, \mu+3 \sigma]\)

Assume that \(P(A \cap B)=0.1, P(A)=0.4\), and \(P\left(A^{c} \cap B^{c}\right)=\) 0.2. Find \(P(B)\).

A patient underwent a diagnostic test for hypothyroidism. The diagnostic test correctly identifies patients who in fact have the disease in \(93 \%\) of the cases and correctly identifies healthy patients in \(81 \%\) of the cases. If 4 in 100 individuals have the disease, what is the probability that a test comes back negative?

Six customers arrive at a bank at the same time. Only one customer at a time can be served. In how many ways can the six customers be served?

Use the random-number generator on a graphing calculator to generate three samples, each of size 10, from a uniform distribution over the interval \((0,1)\). (a) Compute the sample mean and the sample variance of each sample. (b) Combine all three samples, and compute the mean and the sample variance of the combined sample. (c) Compare your answers in (a) and (b) with the true values of the mean and the variance.

Toss a fair coin 200 times. (a) Use the central limit theorem and the histogram correction to find an approximation for the probability that the number of heads is at least 120 . (b) Use Markov's inequality to find an estimate for the event in (a), and compare your estimate with that in (a).

The following table contains the number of flower heads per plant in a sample of size 20 : $$\begin{array}{lllll}15 & 17 & 19 & 18 & 15 \\ 17 & 18 & 15 & 14 & 19 \\ 17 & 15 & 15 & 18 & 19 \\ 20 & 17 & 14 & 17 & 18\end{array}$$ (a) Find the relative frequency distribution. (b) Compute the average value by (i) averaging the values in the table directly and (ii) using the relative frequency distribution obtained in (a).

Assume that a quantitative character is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Determine what fraction of the population falls into the given interval. \([\mu+\sigma, \mu+2 \sigma]\)

Assume that \(P(A)=0.4, P(B)=0.4\), and \(P(A \cup B)=0.7\). Find \(P(A \cap B)\) and \(P\left(A^{c} \cap B^{c}\right)\).

A screening test for a disease shows a positive test result in \(95 \%\) of all cases when the disease is actually present and in \(20 \%\) of all cases when it is not. When the test was administered to a large number of people, \(21.5 \%\) of the results were positive. What is the prevalence of the disease?

An amino acid is encoded by triplet nucleotides. How many different amino acids are possible if there are four different nucleotides that can be chosen for a triplet?

Suppose that \(X\) is exponentially distributed with mean \(1 .\) A computer generates the following sample of independent observations from the population \(X\) : $$ \begin{array}{l} 0.3169,0.5531,2.376,1.150,0.6174 \\ 0.1563,2.936,1.778,0.7357,0.1024 \end{array} $$ Find the sample mean and the sample variance, and compare them with the corresponding population parameters.

Toss a fair coin 300 times. (a) Use the central limit theorem and the histogram correction to find an approximation for the probability that the number of heads is between 140 and 160 . (b) Use Chebyshev's inequality to find an estimate for the event in (a), and compare your estimate with that in (a).

Suppose that the probability mass function of a discrete random variable \(X\) is given by the following table: $$\begin{array}{cc} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-2 & 0.1 \\ -1 & 0.4 \\ 0 & 0.3 \\ 1 & 0.2 \\ \hline \end{array}$$ (a) Find \(E(X)\). (b) Find \(E\left(X^{2}\right)\). (c) Find \(E[X(X-1)]\).

Assume that a quantitative character is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Determine what fraction of the population falls into the given interval. \((-\infty, \mu-2 \sigma]\)

A drawer contains three bags numbered \(1-3\), respectively. Bag 1 contains three blue balls, bag 2 contains four green balls. and bag 3 contains two blue balls and one green ball. You choose one bag at random and take out one ball. Find the probability that the ball is blue.

A bag contains 10 different candy bars. You are allowed to choose \(3 .\) How many choices do you have?

Suppose \(S_{n}\) is binomially distributed with parameters \(n=\) 200 and \(p=0.3 .\) Use the central limit theorem to find an approximation for \(P\left(99 \leq S_{n} \leq 101\right)\) (a) without the histogram correction and (b) with the histogram correction. (c) Use a graphing calculator to compute the exact probabilities, and compare your answers with those in (a) and (b).

Suppose that the probability mass function of a discrete random variable \(X\) is given by the following table: $$\begin{array}{cc} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline 0 & 0.3 \\ 1 & 0.3 \\ 2 & 0.1 \\ 3 & 0.1 \\ 4 & 0.2 \\ \hline \end{array}$$ (a) Find \(E(X)\). (b) Find \(E\left(X^{2}\right)\) (c) Find \(E(2 X-1)\).

A drawer contains six bags numbered \(1-6\), respectively. Bag \(i\) contains \(i\) blue balls and 2 green balls. You roll a fair die and then pick a ball out of the bag with the number shown on the die. What is the probability that the ball is blue?

During International Movie Week, 60 movies are shown. You have time to see 5 movies. How many different plans can you make?

Suppose \(S_{n}\) is binomially distributed with parameters \(n=\) 150 and \(p=0.4\). Use the central limit theorem to find an approximation for \(P\left(S_{n}=60\right)\) (a) without the histogram correction and (b) with the histogram correction. (c) Use a graphing calculator to compute the exact probabilities and compare your answers with those in (a) and (b).

Suppose that the probability mass function of a discrete random variable \(X\) is given by the following table: $$\begin{array}{rc} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-3 & 0.2 \\ -1 & 0.3 \\ 1.5 & 0.4 \\ 2 & 0.1 \\ \hline \end{array}$$ Find the mean, the variance, and the standard deviation of \(X\).

Toss two fair coins and find the probability of at least one head.

21\. You pick 2 cards from a standard deck of 52 cards. Find the probability that the second card is an ace. Compare this with the probability that the first card is an ace.

A committee of 3 people must be formed from a group of 10. How many committees can there be if no specific tasks are assigned to the members?

The following data represent a sample from a normal distribution with mean 0 and variance \(1:\) $$ \begin{array}{l} -0.68,1.22,1.33,-0.84,-0.06 \\ 0.50,0.03,-0.13,-0.29,-0.47 \end{array} $$ Construct a \(95 \%\) confidence interval.

Suppose a genotypic trait is controlled by 80 loci. Each locus, independently of all others, contributes to the genotypic value of the trait either \(+0.3\) with probability \(0.2,-0.1\) with probability \(0.5\), or \(-0.5\) with probability \(0.3\). (a) Find the mean value of the trait. (b) What proportion of the population has a trait value between \(-12\) and \(-7 ?\)

Suppose that the probability mass function of a discrete random variable \(X\) is given by the following table: $$\begin{array}{rc} \hline \boldsymbol{x} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) \\ \hline-1 & 0.1 \\ -0.5 & 0.2 \\ 0.1 & 0.1 \\ 0.5 & 0.25 \\ 1 & 0.35 \\ \hline \end{array}$$ Find the mean, the variance, and the standard deviation of \(X\).

Toss three fair coins and find the probability of no heads.

You pick 3 cards from a standard deck of 52 cards. Find the probability that the third card is an ace. Compare this with the probability that the first card is an ace.

A standard deck contains 52 different cards. In how many ways can you select 5 cards from the deck?

The following data represent a sample from a normal distribution with mean 0 and variance \(1:\) $$ \begin{array}{l} -1.18,0.52,0.36,-0.16,0.92 \\ 0.68,-0.61,-0.54,0.15,1.04 \end{array} $$ Construct a \(95 \%\) confidence interval.

Suppose a genotypic trait is controlled by 90 loci. Each locus, independently of all others, contributes to the genotypic value of the trait either \(1.1\) with probability \(0.7,0.9\) with probability \(0.1\), or \(0.1\) with probability \(0.2\). (a) Find the mean value of the trait. (b) What proportion of the population has a trait value less than \(72 ?\)

Let \(X\) be uniformly distributed on the set $$S=\\{1,2,3, \ldots, 10\\}$$ That is, $$P(X=k)=\frac{1}{10}, \quad k \in S$$ (a) Find \(E(X)\). (b) Find \(\operatorname{var}(X)\).

Toss four fair coins and find the probability of exactly two heads.

Suppose that you have a batch of red-flowering pea plants of which \(40 \%\) are of genotype \(C C\) and \(60 \%\) of genotype \(C c\). You pick one plant at random and cross it with a white-flowering pea plant. Find the probability that the offspring of this crossing will have white flowers.

An urn contains 15 different balls. In how many ways can you select 4 balls without replacement?

Use a graphing calculator to construct a \(95 \%\) confidence interval for a sample of size 30 from a uniform distribution over the interval \((0,1) .\) Take a class poll to determine the percentage of confidence intervals that contain the true mean. Discuss the result in class.

Let \(X\) be uniformly distributed on the set $$S=\\{1,2,3, \ldots, n\\}$$ where \(n\) is a positive integer; that is, $$P(X=k)=\frac{1}{n}, \quad k \in S$$ (a) Find \(E(X)\). (b) Find \(\operatorname{var}(X)\). Hint: Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$ \sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6} $$

Toss four fair coins and find the probability of three or more heads

Suppose that you have a batch of red- and white-flowering pea plants, and suppose also that all three genotypes \(C C, C c\), and \(c c\) are equally represented in the batch. You pick one plant at random and cross it with a white-flowering pea plant. What is the probability that the offspring will have red flowers?

Twelve people wait in front of an elevator that has room for only \(5 .\) Count the number of ways that the first group of people to take the elevator can be chosen.

How often should you toss a coin to be at least \(90 \%\) certain that your estimate of \(P\) (heads) is within \(0.01\) of its true value?

Assume that \(X\) is a discrete random variable with finite range, and set $$p(x)=P(X=x)$$ (a) Show that $$E(a X+b)=\sum_{x}(a x+b) p(x)$$ (b) Use your result in (a) and the rules for finite sums to conclude that $$E(a X+b)=a E(X)+b$$ 26\. Assume that \(X\) is a discrete random variable with finite range, and set $$p(x)=P(X=x)$$ (a) Show that $$\operatorname{var}(a X+b)=a^{2} \sum_{x}[x-E(X)]^{2} p(x)$$ (b) Use your result in (a) and the rules for finite sums to conclude that $$ \operatorname{var}(a X+b)=a^{2} \operatorname{var}(X) $$

Assume that the mathematics score \(X\) on the Scholastic Aptitude Test (SAT) is normally distributed with mean 500 and standard deviation 100 . (a) Find the probability that an individual's score exceeds 700 . (b) Find the math SAT score so that \(10 \%\) of the students who took the test have that score or greater.

Roll a fair die twice and find the probability of at least one \(4 .\)

A bag contains two coins, one fair and the other with two heads. You pick one coin at random and flip it. Find the probability that the outcome is heads.